If Sandy can afford car payments of $$ \$ 270 $$ per month for 4 years, what is the price of a car that she can allord now? Assume an interost rate of 72 percent HW Score: $$ 66.23 \%, 30.8 $$ of 69 points Sandy can afford a car that costs $$ \$ \square $$ or less. (Round to the nearest dollar as needed.)

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We first note that Sandy makes monthly payments of $$ \$270 $$ for 4 years. Our goal is to find the present value (price of the car) that can be afforded now.

1. Define the given values:
   - Monthly payment: $$ P = 270 $$
   - Total number of payments: $$ n = 4 \text{ years} \times 12 \text{ months/year} = 48 $$
   - Annual interest rate: $$ 72\% $$
   - Monthly interest rate: 
     $$
     r = \frac{72\%}{12} = \frac{0.72}{12} = 0.06
     $$

2. The present value of an annuity formula is:
   $$
   PV = P \times \frac{1 - (1+r)^{-n}}{r}
   $$

3. Substitute the given values:
   $$
   PV = 270 \times \frac{1 - (1.06)^{-48}}{0.06}
   $$

4. Compute $$(1.06)^{-48}$$:
   - First, calculate $$(1.06)^{48}$$.
   - Using logarithms or a calculator, one finds approximately:
     $$
     (1.06)^{48} \approx 16.398
     $$
   - Thus,
     $$
     (1.06)^{-48} \approx \frac{1}{16.398} \approx 0.0610
     $$

5. Now plug this value back into the formula:
   $$
   PV = 270 \times \frac{1 - 0.0610}{0.06}
   $$
   Simplify the numerator:
   $$
   1 - 0.0610 = 0.9390
   $$
   So we have:
   $$
   PV = 270 \times \frac{0.9390}{0.06} \approx 270 \times 15.65
   $$

6. Multiply to get the present value:
   $$
   270 \times 15.65 \approx 4225.5
   $$

7. Rounding to the nearest dollar, Sandy can afford a car that costs approximately:
   $$
   \$4226
   $$

Sandy can afford a car that costs $$ \$4226 $$ or less.
Sandy can afford a car priced at \$4,226 or less.

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